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Thread: Orbit of g

  1. #1
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    Orbit of g

    Hi,

    I know that if G acts on set X then the orbit is
    Orbit of g-eq.latex.gif

    But say G=X.

    Does it mean that I put two g's into the above?
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  2. #2
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    Quote Originally Posted by Roland25 View Post
    Hi,

    I know that if G acts on set X then the orbit of x is $\displaystyle \color{red}Gx=\{g*x : \ g \in G \}.$

    But say G=X. Does it mean that I put two g's into the above?
    if $\displaystyle G=X,$ and $\displaystyle x \in X,$ then $\displaystyle \text{orbit}_G(x)=Gx=\{g*x: \ g \in G \}.$ clearly if $\displaystyle *$ is the same as multiplication operation of G, then $\displaystyle Gx=G,$ for all $\displaystyle x \in X=G.$
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  3. #3
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    Sorry, I meant G(x) in the above.
    What I mean is that if Gx={g*x : g in G}
    what is Gg if g is in G.
    Would it just be Gg={G} ?
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  4. #4
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    Quote Originally Posted by Roland25 View Post
    Sorry, I meant G(x) in the above.
    What I mean is that if Gx={g*x : g in G}
    what is Gg if g is in G.
    Would it just be Gg={G} ?
    it depends how you define the group action $\displaystyle *$. if you define that for all $\displaystyle x,g \in G: \ g*x=gx,$ then $\displaystyle Gx=G.$ but if you define $\displaystyle g*x=gxg^{-1},$ for all $\displaystyle g,x \in G,$ then $\displaystyle Gx=\{gxg^{-1}: \ g \in G \},$

    which is the conjugacy class of $\displaystyle x.$
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  5. #5
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    thanks for the help so far.

    I think * was multiplication in my case and so Gx=G kinda seems more appropriate.

    What I still don't get however is what the actual different between Gx and Gg would be if X=G?

    Say X doesn't equal G, would Gx and Gg differ then?

    This is where g is in G
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  6. #6
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    Quote Originally Posted by Roland25 View Post
    thanks for the help so far.

    I think * was multiplication in my case and so Gx=G kinda seems more appropriate.

    What I still don't get however is what the actual different between Gx and Gg would be if X=G?

    Say X doesn't equal G, would Gx and Gg differ then?

    This is where g is in G
    $\displaystyle x \in X$ is fixed and $\displaystyle g$ is any element of $\displaystyle G.$ this should be clear from the definition: $\displaystyle Gx= \{g*x: \ g \in G \}.$
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